Locally Convex Topological Vector Spaces
Since convexity will play a central role in all following chapters, the scalar field K over which vector spaces are defined is from now on assumed to be the real field R or the complex field C, unless the contrary is expressly stated. In most definitions and results (for example, the Hahn-Banach theorem) we shall not find it necessary to distinguish between the real and complex case. When several vector spaces occur in one statement and no explicit mention of the respective scalar fields is made, the spaces involved are assumed to be defined over the same field K, where either K = R or K = C. If K = C, R will be considered a subfield and restriction of scalars to R will be indicated by the use of the adjective “ real ” (Chapter I, Section 7). In particular, the symbols > and ≧, when used between scalars, refer to the customary order in R; for example, “λ > 0” means “λ ∈ R and λ>0”.
KeywordsBanach Space Convex Hull Convex Subset Topological Vector Space Convex Space
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